| L(s) = 1 | − 0.335·2-s + 1.29·3-s − 7.88·4-s + 9.09·5-s − 0.432·6-s − 17.7·7-s + 5.32·8-s − 25.3·9-s − 3.04·10-s + 43.8·11-s − 10.1·12-s + 37.0·13-s + 5.93·14-s + 11.7·15-s + 61.3·16-s + 4.00·17-s + 8.48·18-s − 64.6·19-s − 71.7·20-s − 22.8·21-s − 14.6·22-s − 164.·23-s + 6.87·24-s − 42.2·25-s − 12.4·26-s − 67.5·27-s + 139.·28-s + ⋯ |
| L(s) = 1 | − 0.118·2-s + 0.248·3-s − 0.985·4-s + 0.813·5-s − 0.0294·6-s − 0.956·7-s + 0.235·8-s − 0.938·9-s − 0.0963·10-s + 1.20·11-s − 0.245·12-s + 0.790·13-s + 0.113·14-s + 0.202·15-s + 0.958·16-s + 0.0571·17-s + 0.111·18-s − 0.780·19-s − 0.801·20-s − 0.237·21-s − 0.142·22-s − 1.48·23-s + 0.0584·24-s − 0.338·25-s − 0.0937·26-s − 0.481·27-s + 0.942·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 \) |
| good | 2 | \( 1 + 0.335T + 8T^{2} \) |
| 3 | \( 1 - 1.29T + 27T^{2} \) |
| 5 | \( 1 - 9.09T + 125T^{2} \) |
| 7 | \( 1 + 17.7T + 343T^{2} \) |
| 11 | \( 1 - 43.8T + 1.33e3T^{2} \) |
| 13 | \( 1 - 37.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 4.00T + 4.91e3T^{2} \) |
| 19 | \( 1 + 64.6T + 6.85e3T^{2} \) |
| 23 | \( 1 + 164.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 218.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 252.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 83.8T + 5.06e4T^{2} \) |
| 41 | \( 1 + 95.4T + 6.89e4T^{2} \) |
| 47 | \( 1 + 180.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 401.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 166.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 834.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 651.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 47.7T + 3.57e5T^{2} \) |
| 73 | \( 1 - 763.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 395.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 423.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.22e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 374.T + 9.12e5T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.556203334738190749160935191949, −8.136190634955480139690263672973, −6.49499802728162409944434946622, −6.26783837704939721283738616519, −5.36370973904302312243142739063, −4.16340641172185962429039866807, −3.55035693945404204627210635926, −2.43563967114099631579531117879, −1.17390798058333781028967450915, 0,
1.17390798058333781028967450915, 2.43563967114099631579531117879, 3.55035693945404204627210635926, 4.16340641172185962429039866807, 5.36370973904302312243142739063, 6.26783837704939721283738616519, 6.49499802728162409944434946622, 8.136190634955480139690263672973, 8.556203334738190749160935191949
